Abstract
We consider kernels of unbounded Toeplitz operators in H^p({mathbb {C}}^{+}) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in H^p({mathbb {C}}^{+}), we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of z. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols.
Highlights
In [20], Sarason presented the basic theory of unbounded Toeplitz operators in H2(D) with symbols in L2(T) and, motivated by natural questions that lead to other types of symbols [13,22,23], of Toeplitz operators with analytic and co-analytic symbols in more general classes
We would like to examine their kernels and study what properties are shared with kernels of bounded Toeplitz operators, which have attracted great interest for their rich structure and the information that they provide on the corresponding Toeplitz operators
In the class of bounded Toeplitz operators, for each φ+ ∈ Hp+\{0} one can find a minimal kernel Kmin(φ+), which is contained in any other Toeplitz kernel to which φ+ belongs
Summary
In [20], Sarason presented the basic theory of unbounded Toeplitz operators in H2(D) with symbols in L2(T) and, motivated by natural questions that lead to other types of symbols [13,22,23], of Toeplitz operators with analytic and co-analytic symbols in more general classes. Unbounded Toeplitz operators appear naturally, for instance when studying inverses or generalized inverses of Toeplitz operators with bounded symbols [10,18]. The inverse of a bounded Toeplitz operator, if it exists, is the composition of two Toeplitz operators which, in general, are unbounded. Toeplitz operators in the Hardy space H2(C+) of the upper half-plane C+, and more generally, in Hp(C+), p > 1, which arise in many applications [8,15,18,21,24,25,26].
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